--- title: Discrete Virtual Holonomic Constraints url: https://www.emergentmind.com/topics/discrete-virtual-holonomic-constraints-dvhc type: topic --- # Discrete Virtual Holonomic Constraints Searching arXiv for recent papers on Discrete Virtual Holonomic Constraints to ground the article in current literature. Discrete Virtual Holonomic Constraints (DVHC) are algebraic constraints enforced only at discrete instants of a hybrid system, rather than continuously in time. In the devil-stick problems studied in 2025, a DVHC specifies the center-of-mass (CoM) location as a function of the stick orientation at the instants when impulsive inputs are applied, and this construction induces a discrete zero dynamics (DZD) that governs orbital behavior. In cooperative quadrupedal locomotion, closely related discrete virtual constraints appear as modified local outputs designed so that an augmented periodic orbit remains invariant under both continuous and discrete dynamics. Across these settings, DVHC provides a framework for geometric constraint specification, hybrid-order reduction, and orbit stabilization through return-map methods [2508.15040][2509.08085][1902.03690]. ## 1. Formal definition and basic interpretation A continuous virtual holonomic constraint is an algebraic relation \(h(q)=0\) in the generalized coordinates \(q\), enforced for all \(t\) by continuous-time feedback. By contrast, a DVHC is enforced only at the discrete instants \(t_k\) when impulses are applied [2508.15040]. In the propeller-motion formulation for a devil-stick, the generalized coordinates immediately before the \(k\)-th impulse are \[ q_k^- := [h_k^-,\theta_k]^T, \] with \(h_k^- \in \mathbb{R}^2\) the CoM position and \(\theta_k\) the orientation. The DVHC is \[ \rho_k := h_d(q_k^-) = h_k^- - \Phi(\theta_k)=0, \] where \(\Phi:\mathbb{R}\to\mathbb{R}^2\) is a prescribed virtual trajectory, for example \[ \Phi(\theta)= [R\cos(\theta-\phi),\,R\sin(\theta-\phi)]^T. \] Thus, at each impulse instant the condition \(h_k^-=\Phi(\theta_k)\) is re-imposed, but between impulses \(h_k\) need not lie on the circle [2508.15040]. The planar-juggling formulation uses the same principle with a different geometry. If the pre-impact state is \[ x_k=(h_x(k),\,h_y(k),\,\theta_k), \] then the DVHC is \[ h_d(x_k):=h(k)-\Phi(\theta_k)\in\mathbb{R}^2, \qquad \Phi(\theta)=\begin{bmatrix}\alpha\tan\theta\\ \beta\end{bmatrix}, \quad \alpha>0,\ \beta>0. \] Its stated interpretation is that, at each impact, the CoM is “snapped” to the point \((\alpha\tan\theta_k,\beta)\) via an impulsive control [2509.08085]. In cooperative locomotion, the discrete-constraint idea is embedded in distributed virtual-output design. Each agent \(i\in\{1,2\}\) implements modified virtual outputs \[ y^i_{v,w}(t,x_i,\Theta,\xi_i) = y_v(t,x_i) - \begin{bmatrix} \alpha_{v,w}(s_j-s^*(\tau,w))\\ C_{v,w}^{roll}(q_j^{roll}-q^{roll*}(\tau,w))\\ C_{v,w}^{pitch}(q_j^{pitch}-q^{pitch*}(\tau,w)) \end{bmatrix}, \] and these are required to vanish on the augmented orbit \(\mathcal{C}_d^a\). In that setting, DVHC are enforced through local distributed feedback laws rather than through a single impulsive law [1902.03690]. A recurring point in the literature is that DVHC are geometric rather than time-parametrized prescriptions. One stated consequence is that they obviate the need for a time-varying reference trajectory in the devil-stick setting [2509.08085]. ## 2. Hybrid formulation and discrete zero dynamics The devil-stick problems are modeled as hybrid systems with alternating impulse and flight phases. In the propeller-motion formulation, an impulse \(I\) is applied normal to the stick at signed distance \(r_k\), and the state then evolves under free fall for duration \(\delta_k\) until the next impulse [2508.15040]. The impulsive update is \[ h_k^+ = h_k^-, \qquad \theta_k^+ = \theta_k^-=\theta_k, \] \[ v_k^+ = v_k^- + \frac{1}{m}I[-\sin\theta_k,\cos\theta_k]^T, \qquad \omega_k^+ = \omega_k^- + \frac{I r_k}{J}. \] The subsequent flight dynamics are \[ h_{k+1}^- = h_k^+ + v_k^+\delta_k - \frac{1}{2}[0,g\delta_k^2]^T, \] \[ v_{k+1}^- = v_k^+ + [0,-g]^T\delta_k, \qquad \theta_{k+1}=\theta_k+\omega_k^+\delta_k, \qquad \omega_{k+1}^-=\omega_k^+. \] Because the next impulse is applied after a prescribed rotation increment \(\Delta\theta\), one enforces \[ \theta_{k+1}^- - \theta_k^+ = \Delta\theta, \] which yields \(\delta_k=\Delta\theta/\omega_{k+1}\) and \(\omega_{k+1}=\omega_k+I r_k/J\) [2508.15040]. Under the DVHC, the pre-impact position and velocity are constrained by \[ h_k^- \equiv \Phi(\theta_k), \qquad v_k^- \equiv \Psi(\theta_k,\omega_k), \] where \[ \Psi(\theta,\omega)= \frac{(\omega\Delta\theta)[\Phi(\theta)-\Phi(\theta-\Delta\theta)]}{\Delta\theta} - \begin{bmatrix} 0\\ g/(2\omega) \end{bmatrix}. \] Substitution into the hybrid update yields a two-dimensional DZD in \((\theta_k,\omega_k)\) alone [2508.15040]. For propeller motion, the implicit return equations are \[ \theta_{k+1}-\theta_k-\Delta\theta=0, \] \[ S_m\,\omega_{k+1}-S_p\,\omega_k+K\sin\theta_k\left(\frac{1}{\omega_{k+1}}+\frac{1}{\omega_k}\right)=0, \] with \[ S_p=\sin(\phi+\Delta\theta/2),\qquad S_m=\sin(\phi-\Delta\theta/2),\qquad K=\frac{g^2}{4R\sin(\Delta\theta/2)}. \] An explicit quadratic solution for \(\omega_{k+1}\) is also given in the source [2508.15040]. The planar-juggling formulation arrives at the same dimensional reduction in a different coordinate chart. Enforcing the DVHC exactly, with \(\rho_k\equiv 0\) in positions and \(D_k\equiv 0\) in velocities, yields a reduced map on \[ \zeta_k := \begin{bmatrix}\theta_k\\ \omega_k\end{bmatrix}\in S^1\times\mathbb{R}, \] so that the 4-dimensional hybrid underactuated system reduces to a 2-dimensional discrete dynamics when the DVHC is enforced exactly [2509.08085]. In cooperative locomotion, the hybrid setting is substantially larger. The single-agent model is \[ \Sigma=(G,D,S,FG,\Delta), \] with \(G=(V,E)\) an 8-vertex directed cycle for Vision 60, and the two-agent model \(\Sigma^a\) is constructed on the strong-product graph \(G^a=G\boxtimes G\). Eliminating multipliers yields impact maps of the form \[ x^{a+}=\Delta_{(v,w)\to(v',w')}^a(x^{a-}), \] and the design objective is that the augmented orbit \[ \mathcal{C}_d^a := \{x_1=\phi^*(t),\ x_2=T_d(\phi^*(t))\ |\ 0\le t0\), then \(I_k\) is recovered, and finally \[ r_k=\frac{J}{I_k\delta_k}-\frac{J\omega_k}{I_k}. \] The source states that as \(\rho_k\to 0\), the velocity error \(D_k\to 0\), so the DVHC manifold is reached and maintained at the impulse instants [2508.15040]. The planar-juggling controller uses the same structure, with \[ \rho_k := h(k)-\Phi(\theta_k), \qquad D_k := v(k)-\Psi(\theta_k,\omega_k), \] and the exponential law \[ \rho_{k+1}=\Lambda \rho_k, \qquad \Lambda=\operatorname{diag}(\lambda_x,\lambda_y), \qquad 0\le \lambda_i<1. \] On the zero dynamics, closed-form expressions are reported: \[ \delta_k=\frac{(-1)^k2\alpha\omega_k}{g}\left(1-\frac{\tan\theta_{k+1}}{\tan\theta_k}\right), \] \[ I_k=\frac{(-1)^k m}{\cos\theta_k}\left[\alpha\omega_k\left(1-\frac{\tan\theta_{k+1}}{\tan\theta_k}\right)+\frac{g}{2\omega_k}\right], \] \[ r_k=\frac{(-1)^{k+1}J\cos\theta_k}{m\alpha\left(1-\frac{\tan\theta_{k+1}}{\tan\theta_k}\right)}. \] These expressions are specific to the planar-juggling geometry \(\Phi(\theta)=[\alpha\tan\theta,\beta]^T\) [2509.08085]. The orbit-stabilizing layer in both devil-stick papers is formulated through an Impulse-Controlled Poincaré Map. A section \(\Sigma\) is chosen, the return state \(z\) is defined on successive intersections, and the map \[ z(j+1)=P[z(j),I(j),r(j)] \] is linearized about a fixed point: \[ e(j+1)=A\,e(j)+B\,u(j), \qquad u(j)=[I-I^*,\,r-r^*]^T. \] If \((A,B)\) is stabilizable or controllable, a discrete feedback \[ u(j)=K\,e(j) \] is selected so that the eigenvalues of \(A+BK\) lie inside the unit circle. In both sources, one stated choice is LQR with \(Q=I_5\) and \(R=2I_2\) [2508.15040][2509.08085]. In cooperative locomotion, control synthesis is distributed and continuous between impacts. Each agent solves, at each 1 kHz sample, a local quadratic program \[ \min_{u_i,\delta}\ \frac12\|u_i-\Gamma_v^{nom}(t,x_i)\|^2+\frac{w}{2}\|\delta\|^2 \] subject to \[ A_{v,w}^i u_i+b_{v,w}^i+\delta=-e_{v,w}^i, \qquad u_{min}\le u_i\le u_{max}, \qquad \delta_{min}\le \delta\le \delta_{max}. \] The cost keeps \(u_i\) close to the nominal HZD law \(\Gamma_v^{nom}\), while \(\delta\) penalizes input-output linearization defect [1902.03690]. ## 5. Demonstrated systems and simulation evidence The propeller-motion study reports four classes of simulation evidence [2508.15040]. For DVHC enforcement without orbit stabilization, the parameters are \[ m=0.1\ \text{kg},\quad \ell=0.5\ \text{m},\quad J=\frac{1}{12}m\ell^2,\quad R=1\ \text{m},\quad \phi=\frac{\pi}{2},\quad \Delta\theta=\frac{2\pi}{6},\quad \lambda_x=\lambda_y=0.5. \] From arbitrary \(q_1\), \(\rho_1\neq 0\), and \(D_1\neq 0\), the reported behavior is \(\rho_k\to 0\) exponentially and \(D_k\to 0\); \(I_k\) and \(r_k\) converge to periodic sequences; \((\theta_k,\omega_k)\) settles into a one-parameter family of orbits; and an approximate invariant \(\bar E_k\) flattens out. On the Poincaré section \(\Sigma:\theta=\pi/3\), a desired orbit with \(\omega^*=8\ \text{rad/s}\) and LQR weights \(Q=I_5\), \(R=2I_2\) yields convergence to a single orbit, with impulse and lever-arm corrections vanishing after \(j\approx 3\). Additional simulations show aperiodic DVHC motion when \(\phi\neq \pm \pi/2\), exact \(N\)-periodicity for certain rational choices of \(\Delta\theta\), and equidistribution of \(\theta_k\) on \(S^1\) for irrational \(\Delta\theta\) [2508.15040]. The planar-juggling study reports a separate parameter set: \[ m=0.1\ \text{kg},\quad \ell=0.5\ \text{m},\quad J=0.0021\ \text{kg}\cdot\text{m}^2,\quad g=9.81\ \text{m/s}^2, \] \[ \alpha=0.6131,\quad \beta=3,\quad \text{odd}=\frac{\pi}{6},\quad \text{even}=\frac{5\pi}{6},\quad \lambda_x=\lambda_y=0.5. \] From arbitrary \((h,v,\theta,\omega)\) initial conditions, the errors \(\|\rho_k\|\) and \(\|D_k\|\) decay \(\approx 0.5\times\) per impact. The resulting 2-periodic orbit has \[ \omega_k \approx \{-3.1596\ \text{rad/s}\ \text{(odd)},\ +5.5532\ \text{rad/s}\ \text{(even)}\}, \] \[ \delta_k \approx \{0.3771\ \text{s},\ 0.6629\ \text{s}\}, \qquad I_k = (-1)^k\,0.5890\ \text{Ns}, \qquad r_k=0.0308\ \text{m}. \] For orbital stabilization, the study chooses \(\omega^*=-4.1888\ \text{rad/s}\) in the planar-symmetric case, with \(\delta_k=0.5\ \text{s}\), and reports convergence of \(\omega_k\to \omega^*\) and \(\delta_k\to 0.5\ \text{s}\) in \(O(10)\) impacts [2509.08085]. The cooperative-locomotion study addresses a much larger hybrid model. Its numerical model has 64 continuous-time domains, 192 discrete-time transitions, 96 state variables, and 36 control inputs. The implementation is on two Vision 60 quadrupeds with Kinova arms, and the reported outcome is that the distributed QP-based controllers validate the effectiveness of DVHC and the associated distributed feedback design [1902.03690]. Taken together, these examples show that DVHC have been used in both low-dimensional underactuated juggling and high-dimensional cooperative legged locomotion. A plausible implication is that the framework is not tied to a single morphology, but to hybrid systems in which actuation, switching, or impact structure makes event-based manifold enforcement natural. ## 6. Relation to continuous VHCs, HZD, and conceptual scope The relation between DVHC and continuous VHCs is explicit in the propeller-motion paper. As \(\Delta\theta\to 0\), one has \(\delta_k\to 0\) and \[ \frac{\theta_{k+1}-\theta_k}{\delta_k}\to \omega, \qquad \frac{\omega_{k+1}-\omega_k}{\delta_k}\to \dot\omega. \] Expanding the discrete dynamics in this limit recovers the zero dynamics of continuous propeller motion under a VHC: \[ \ddot\theta= -\left(\frac{g}{R}\sin\phi\right)\sin\theta+\cot\phi\,\dot\theta^2. \] For \(\phi=\pm \pi/2\), this reduces to \[ \ddot\theta=\left(\pm \frac{g}{R}\right)\sin\theta, \] the planar pendulum-type zero dynamics whose periodic orbits govern continuous propeller maneuvers [2508.15040]. This establishes DVHC as a discrete analogue of continuous VHCs in a precise limiting sense. The cooperative-locomotion construction connects DVHC to hybrid zero dynamics (HZD). For a single agent, the nominal outputs \[ y_v(t,x)= \begin{bmatrix} s(q,\dot q)-s^*(\tau,v)\\ C_v(q-q^*(\tau,v)) \end{bmatrix}=0 \] are designed so that \[ HZD_v=\{y_v=0,\ \dot y_v^h=0\} \] is hybrid-invariant and yields a stable periodic orbit \(\mathcal{C}\). The distributed extension modifies these outputs through measurable global variables \(\Theta\), so that on the augmented cooperative orbit \(\mathcal{C}_d^a\), each agent applies exactly the nominal single-agent controller. By Theorem 2 and Corollary 5 in the source, \(\mathcal{C}_d^a\) is then a periodic orbit of the complex hybrid model \(\Sigma^a\) [1902.03690]. Two misconceptions are directly addressed by the reported results. First, DVHC are not continuous path constraints: in the devil-stick setting, the CoM is constrained only at impulse instants, and between impulses it may depart from the prescribed geometric locus [2508.15040]. Second, exact constraint enforcement is not identical to orbital stabilization: additional return-map feedback is used precisely because the induced DZD can admit families of periodic motions, slow drift, or aperiodic behavior, depending on parameter choices [2508.15040][2509.08085]. The literature therefore presents DVHC as a hybrid-systems methodology in which one prescribes geometry at discrete events, derives a low-dimensional DZD, and then designs event-based or distributed feedback to stabilize desired orbits. This suggests a unifying interpretation of DVHC as event-centered virtual-constraint design for underactuated and contact-rich robotic systems. Source: https://www.emergentmind.com/topics/discrete-virtual-holonomic-constraints-dvhc